They will, but those not eligible will just get an automatic passport.

They need that system anyway for babies too young to be vaccinated, people with health issues making the vaccine inadvisable, etc.

Who is "they"? In the US it will mostly be individual businesses making whatever decisions make the most sense from a business perspective.

UK, as was discussed in the thread above

Presumably the UK where there is lots of discussion about the government possibly making vaccine passports mandatory for certain types of businesses (I think large-scale events and nightclubs are examples).

There's been some talk in the US at the state level (esp. NY) but AFAIK the federal government has explicitly said they're not going to be getting into vaccine passports.

These are independent things in bureaucracy. Here in Canada they are starting to talk about passports while vaccinations are under 2%